II. Write as many 9's as there are figures in the repetend, with as many ciphers annexed as there are finite decimal figures, for the required denominator. EXAMPLES FOR PRACTICE. 1. Reduce .57 to a common fraction. 2. Reduce .048 to a common fraction. 3. Reduce .6472 to a common fraction. 4. Reduce .6590 to a common fraction. 5. Reduce .04648 to a common fraction. 6. Reduce .1004 to a common fraction. 7. Reduce .9285714 to a common fraction. 8. Reduce 5.27 to a common fraction. 9. Reduce 7.0126 to a mixed number. 10. Reduce 1.58231707 to an improper fraction. 11. Reduce 2.029268 to an improper fraction. CASE III. 240. To make two or more repetends similar and conterminous. 1. Make .47, .53675, and .37234 similar and conterminous. OPERATION. 1234 66 7 8 9 9 8 S 47 = .47474747474747 .53675.58675675675675 Ans. .37234.37234723472347 ANALYSIS. The first of the given repetends begins at the place of tenths, the second at the place of thousandths, and the third at the place of hundredths; and as the points in any repetend cannot be moved to the left over the finite places, we can make the given repetends similar, only by moving the points of at least two of them to the right. Again, the first repetend has 2 places, the second 3 places, and the third 4 places; and the number of places in the new repetends must be at least 12, which is the least common multiple of 2, 3, and 4.. We therefore expand the given repetends, and place the first point in each new repetend over the third place in the decimal, and the second point over the fourteenth, and thus render them similar and conterminous. Hence the following RULE. I. Expand the repetends, and place the first point in each over the same order in the decimal. II. Place the second point so that each new repetend shall contain as many places as there are units in the least common multiple of the number of places in the several given repetends. NOTE. Since none of the points can be carried to the left, some of them must be carried to the right, so that each repetend shall have at least as many finite places as the greatest number in any of the given repetends. EXAMPLES FOR PRACTICE. 1. Make .43, .57, .4567, and .5037 similar and conterminous. 2. Make .578, .37, .2485, and 04 similar and conterminous. 3. Make 1.34, 4.56, and .341 similar and conterminous. 4. Make .5674, .34, .247, and .67 similar and conterminous. 5. Make 1.24, .0578, .4, and .4732147 similar and conter minous. 6. Make .7, .4567, .24, and .346789 similar and conterminous. 7. Make .8, 36, .4857, .34567, and .2784678943 similar and conterminous. ADDITION AND SUBTRACTION. 241. The processes of adding and subtracting circulating decimals depend upon the following properties of repetends: I. If two or more repetends are similar and conterminous, their denominators will consist of the same number of 9's, with the same number of ciphers annexed. Hence, II. Similar and conterminous repetends have the same denominators and consequently the same fractional unit. 1. Add .51, 3.24 and, 2.785. OPERATION. = .54 .54444 3.24 3.24242 2.785 = 2.78527 6.57214 ANALYSIS. Since fractions can be added only when they have the same fractional unit, we first make the repetends of the given decimals similar and conterminous. We then add as in finite decimals, observing, however, that the 1 which we carry from the left hand column of the repetends, must also be added to the right hand column; for this would be required if the repetends were further expanded before adding. 2. From 7.4 take 2. 7852. OPERATION. 7.4444 2.7852 4.658i ANALYSIS. Since one fraction can be subtracted from another only when they have the same fractional unit, we first make the repetends of the given decimals similar and conterminous. We then subtract as in finite decimals; observing that if both repetends were expanded, the next figure in the subtrahend would be 8, and the next in the minuend 4; and the subtraction in this form would require 1 to be carried to the 2, giving 1 for the right hand figure in the remainder. 242. From these principles and illustrations we derive the following RULE. I. When necessary, make the repetends similar and con terminous. II. To add ;-Proceed as in finite decimals, observing to increase the sum of the right hand column by as many units as are carried from the left hand column of the repetends. III. To subtract;— Proceed as in finite decimals, observing to diminish the right hand figure of the remainder by 1, when the repetend in the subtrahend is greater than the repetend of the minuend. IV. Place the points in the result directly under the points above. NOTE. When the sum or difference is required in the form of a common fraction, proceed according to the rule, and reduce the result. EXAMPLES FOR PRACTICE. 1. What is the sum of 2.4, .32, .567, 7.056, and 4.37 ? Ans. 14.7695877. 2. What is the sum of .478, .321, .78564, .32, .5, and .4326 ? 3. From 7854 subtract .59. Ans. 2.8961788070698. 5. What is the sum of .5, .32, and .12 ? Ans. 29.7455. Ans. 1. 6. What is the sum of .4387, .863, .21, and .3551? 8. From 432 subtract .25. 9. From 7.24574 subtract 2.634. Ans. .18243. Ans. 4.61. 10. From .99 subtract .433. Ans. .55656. 11. What is the sum of 4.638, 8.318, .016, .54, and .45? 12. From .4 subtract .23. MULTIPLICATION AND DIVISION. 243. 1. Multiply 2.428571 by .063. Ans. 133. ANALYSIS. We first reduce the multiplicand and multiplier to their equivalent fractions, and obtain 47 and 170; then 47 × 10 ==.154. 244. From these illustrations we have the following RULE. Reduce the given numbers to common fractions; then multiply or divide, and reduce the result to a decimal. UNITED STATES MONEY. 245. By Act of Congress of August 8, 1786, the dollar was declared to be the unit of Federal or United States Money; and the subdivisions and multiples of this unit and their denominations, as then established, are as shown in the 246. By examining this table we find 1st. That the denominations increase and decrease in a tenfold ratio. 2d. That the dollar being the unit, dimes, cents and mills are respectively tenths, hundredths and thousandths of a doilar. 3d. That the denominations of United States money increase and decrease the same as simple numbers and decimals. Hence we conclude that I. United States money may be expressed according to the deci mal system of notation. II. United States money may be added, subtracted, multiplied and divided in the same manner as decimals. NOTATION AND NUMERATION. 247. The character $ before any number indicates that it expresses United States money. Thus $75 expresses 75 dollars. 248. Since the dollar is the unit, and dimes, cents and mills are tenths, hundredths and thousandths of a dollar, the decimal point or separatrix must always be placed before dimes. Hence, in any number expressing United States money, the first figure at the right of the decimal point is dimes, the second figure is cents, the third figure is mills, and if there are others, they are tenthousandths, hundred-thousandths, etc., of a dollar. Thus, $8.3125 |